Our defence of CSE echoes the elaborations of a small but grow-ing literature on equality through time, includgrow-ing Gosseries (2014), Bidadanure (2016), and Daniels (1988, 1993, 2008). In this section, we briefly discuss their proposals and compare them to our approach.
Like us, these authors argue that CLE is the essential build-ing block of any intertemporal egalitarian approach, even though it should be supplemented with additional restrictions to deal with ‘changing places egalitarianism’. Gosseries (2014) has proposed a sufficiency restriction on CLE, according to which at every point in time along the life of a person, each should have enough to cover at least their basic needs. For him, if people are equal over their complete lives but fall be-low a given welfare threshold at some point in their lives, the demands of temporal justice are not met. This approach likely captures some of our intuitions in rejecting extreme simultane-ous inequalities among agents, but it does not properly address the issues raised by changing places egalitarianism in scenarios with less extreme, but still morally relevant inequalities. This is not surprising because, unlike our corresponding segments restriction, sufficientarianism is not meant to capture egalitar-ian intuitions. Indeed, sufficientaregalitar-ianism has explicitly been proposed as an alternative to egalitarianism and embodies the intuition that “equality is not, as such, of particular moral im-portance” (Frankfurt , 1987, 21).
According to Bidadanure (2016), the strong dystopian feel-ing created by extreme examples of changfeel-ing places egalitar-ianism cannot really be captured in a standard welfarist,
dis-tributive framework. She argues against SSE and claims that by moving away from the paradigm of distributive egalitarian-ism we can find a non-arbitrary complement to CLE focusing on a relational perspective, according to which people should be treated as equals.27 She therefore endorses a relational egal-itarian complement to CLE which limits the scope of acceptable synchronic inequalities.
The relational approach proposed by Bidadanure is inter-esting and innovative, and it may be an essential component of a complex, multifaceted approach to egalitarianism. As she aptly notes, distributive and relational approaches complement each other as they “simply appeal to different kinds of reasons to care about inequalities” (Bidadanure, 2016, 238). Nonethe-less, an emphasis on the relational dimension of egalitarianism does not provide a complete answer to the equally important distributive questions. Indeed, her approach leaves the central questions raised in the literature on the distributive dimension of temporal inequalities largely unanswered. While acknowl-edging the relevance of relational considerations for egalitari-anism, our paper focuses precisely on the appropriate distribu-tive benchmark.
In a series of seminal contributions, Daniels (1988, 1993, 2008) has argued that both lifetimes and temporal stages of lives should be taken into account within a Prudential Lifes-pan Approach. According to him, complete lives remain the key unit of egalitarian concern but CLE should be constrained by a prudential procedure to allocate resources between young and old. To be precise, in the Prudential Lifespan Approach, the best way of synchronically distributing resources among peo-ple of different ages should be identified by thinking pruden-tially about a diachronic distribution across the different tem-poral stages of a single complete lifetime. As a result, there may be significant simultaneous inequalities, but this is “not by itself a form of age-bias. This differential treatment would not be morally objectionable, . . . , if it made each life go as well as possible (it was a ‘prudent’ allocation) and if all people were treated similarly over their whole lives” (Daniels, 2008, 483).
27 See, for example, Anderson (1999).
Our results in section 2 are consistent with Daniels’ (1988;
1993; 2008) account of intertemporal justice: under the assump-tions of our model, the prudential lifetime allocation will coin-cide with the maximin solution. More generally, if agents are fundamentally alike, and the economic environment does not undergo major structural changes, then the prudential lifespan approach will yield a CS-egalitarian allocation. Nonetheless, some important differences should be noted. First, Daniels’ can be considered as a mixed account, which imposes a prudential
—and thereby non-egalitarian—constraint on CLE. In contrast, in our account, CSE emerges as the appropriate intertemporal benchmark, based on purely egalitarian considerations. Max-imin, or prudential considerations are clearly distinguished, and theoretically subsidiary in our analysis, and therefore—
unlike in Daniels’ (1988; 1993; 2008) account—the defence of CSE is independent on the specific formulation of maximin, or prudential accounts.28 In fact, formally, under our assump-tions, while CSE identifies a set of egalitarian allocaassump-tions, the prudential lifespan account picks up one (or a strict subset) of such allocations. Second, in our model, we do not assume that the agents, or the social planner, actually act prudentially subject to a CLE constraint: the fact that the allocation in The-orem 1 corresponds with the allocation advocated by the pru-dential lifetime account is a result of a more basic—and norma-tively well founded, at least within an egalitarian approach—
principle, namely Rawls’s maximin.
In addition to the specific points mentioned above, it is worth briefly mentioning two more general methodological fea-tures that differentiate our paper from the literature reviewed here. First, we clearly distinguish the identification of the ap-propriate egalitarian benchmark from the choice of a suitable inequality measure. As we have argued in Section 1, this dis-tinction is important and it is not always properly spelled out.
Second, we clearly distinguish egalitarian and non-egalitarian concerns. Our defence of CSE lies entirely on egalitarian prin-ciples and intuitions, and although we do bring non-egalitarian considerations to bear in Section 2, they are only meant to
pro-28 For a thorough critique, see McKerlie (2012).
vide additional support for CSE.
4 Conclusion
In this paper three egalitarian views are analysed in the in-tertemporal context. Once the static setting is abandoned, egal-itarian principles—apart from differing in the analysis of ex-isting inequalities,—also define different ideal egalitarian dis-tributions. While it may be important to use the different in-formation conveyed by every criterion in the analysis of exist-ing inequalities, when the egalitarian distributions associated with them are analysed, CLE and SSE have undesirable fea-tures while CSE represents the appropriate egalitarian bench-mark.
The relations between the three egalitarian principles and other moral ideals, namely maximin and utilitarianism, are also analysed. As regards the maximin principle, Propositions 1-2 and Theorem 1 show that, unlike with CLE and CSE, the adop-tion of SSE implies a trade off between egalitarianism and a concern for the worst off. As regards utility, the same conclu-sion holds if one interprets SSE as a restriction on CLE, since it yields a lower egalitarian lifetime welfare level. This is not true if SSE is analysed per se, but this is just because in this case the SSE is a strictly intratemporal principle.
In closing this paper, it is worth noting that our formal anal-ysis yields some interesting insights on a vexed issue in nor-mative economics, namely the well-known trade-off between equality and growth. Arrow (1973) and Dasgupta (1974) proved that if agents are selfish, live for one period, and their lives do not overlap, then Rawls’ maximin principle implies a stationary path of consumption, capital and welfare. Our model yields a more nuanced conclusion and suggests some interesting directions for further research. In our framework with overlapping generations, at the maximin solution the path of capital must be chosen so as to maximise lifetime welfare and, under quite general assumptions, this implies growth in at least one period. In other words, although the application
of the maximin principle precludes permanent growth in the economy, as in Arrow (1973) and Dasgupta (1974), the dynam-ics of the economy is not completely stationary.
Our conjecture is that an explicit and more realistic analy-sis of the temporal structure of agents’ lives (which span over many periods), and the overlaps across generations, together with the introduction of uncertainty and irreversibility of in-vestments may alter the justice/growth trade-off.29 This in-dicates a promising line for further research on intertemporal and intergenerational justice.
Appendix
For any variable z, let dz = z′− z denote a change in z.
Proof of Proposition 1.
Let W∗ be the value of MP and suppose that, contrary to the statement, W (c01, c12) > W∗. By continuity, there is a sufficiently small dc01 < 0, such that W (c′01, c12) > W∗, −dk1 = dc01 and the amount of resources available in t = 1 increases by [1 + f′(k1)]dk1. Let dc11 = f′(k1)dk1 > 0 and dk2 = dk1 and repeat the procedure for all t ≥ 2 so that dct1 = f′(kt)dkt > 0, dkt+1 = dkt> 0, and W (c′t1, ct+12 ) > W∗, all t, a contradiction. The proof of the case with W (ct1, ct+12 ) > W∗, some t > 0, is similar.
Proof of Proposition 2.
Suppose not. Then there is dct1, dct+12 such that dct+12 = −[1 + f′(kt+1)]dct1and u′(ct1)dct1+ βu′(ct+12 )dct+12 > 0. By the concavity of W , this implies W (c′t1, c′t+12 ) > W (ct1, ct+12 ) leaving unmodi-fied cj1, all j 6= t and kj, cj2, all j 6= t + 1, violating Proposition 1.
Proof of Lemma 1.
Consider c > cm. At t = 0, k1 < k0and thus k2− k1 = k1− k0+ f (k1)−f (k0) < 0 and k2−k1 < k1−k0, i.e. |k2−k1|/|k1−k0| > 1,
29 A similar point is made by Silvestre (2002), albeit in a rather different formal setting.
and, by induction |kt+1 − kt|/|kt− kt−1| > 1. Therefore kt = 0 for t finite, and c is not sustainable.
Proof of Theorem 1.
1. The existence and uniqueness of (c∗1, c∗2, k∗) is guaranteed by the assumptions on u and f . Note also that (c∗1, c∗2, k∗) satisfies the condition in Propositions 1 and 2.
2. Suppose it is possible to raise the welfare of all genera-tions above W∗. Consider P0: by construction the first generation’s welfare can increase over W∗ if and only if R1 < R0. Consider now generation 2: clearly V (R1, R0) <
W∗. Moreover V (Rt, Rt+1) is concave and its iso- welfare contours have slope [1+f′(k(Rt, Rt+1))], where k(Rt, Rt+1) is the optimum value of kt+1 from Pt. Hence, W (c11, c22) >
W∗implies R2 < R0, with |R2−R0| > [1+f′(k(R0, R0))]|R1− R0|. Iterating the argument, W (ct1, ct+12 ) > W∗ implies
|Rt+1 − R0|/|Rt− R0| > [1 + f′(k(R0, R0))], all t, and the path violates the non- negativity of Rtat some finite t.
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